Unformatted text preview: (go over
the intuition once and for all):
Contrast
The following graphs are for the problem:
In each case, look at change in with
Lagrangian
method due to an infinitesimally small increase in 8
ECO 204 Chapter 3: Practice Problems & Solutions for Utility Maximization in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. {( (3.3) Show if the following utility functions, defined over
(a) ( ) and ) }, have identical indifference curves: ? Show all calculations. Answer
(
). As such, the both
True. The utility function
is a positive monotonic transformation of
represent the same preferences and graphically will have the same indifference curves (albeit with differing numbers)
 (why not check if this is true?). Here is the positive monotonic transformation:
and identical 
( ) Notice that when we do positive monotonic transformations we leave alone the term
microeconomics).
(b) ( ) and ? Show all calculations. False. The utility function is not a positive monotonic transformation of
( This is not the same as (it be the Chuck Norris of ( ): ) . (3.4) Graph an indifference curve for an arbitrary level of utility
● give expressions for the indifference curve
equation, intercepts and slope ● and state (but do not solve) as simplified as possible UMP. Show all calculations.
( C) . [Hint: Do a positive monotonic transformation] Answer:
Do a positive monotonic transformation so we can work with an easier utility function: 9
ECO 204 Chapter 3: Practice Problems & Solutions for Utility Maximization in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. The equation of an indifference curve corresponding to an arbitrary level of utility For the yaxis intercept set However, is: : undefined which means there is no yaxis intercept. For the xaxis intercept set : The slope of the indifference at any bundle in the consumption set is: When
slope is Notice that (yaxis “intercept” – using this word carelessly) the slope is and when (xaxis intercept) the Thus, the indifference curve looks like: whenever and Hence the UMP can be written as:
10 ECO 204 Chapter 3: Practice Problems & Solutions for Utility Maximization in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. (d) . Answer:
The equation of an indifference curve corresponding to an arbitrary level of utility ( For the yaxis intercept set ) : For the xaxis intercept set is: : The slope of the indifference at any bundle in the consumption set is: When (yaxis “intercept”) the slope is and when (xaxis intercept) the slope is Thus, the indifference curve looks like: 11
ECO 204 Chapter 3: Practice Problems & Solutions for Utility Maximization in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. Notice that whenever and Hence the UMP can be written as: (3.5) A consumer has the following utility function defined over Suppose and that {( ) }: and Without solving the UMP, what is impact on the consumer’s utility due to...
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 Fall '09
 AJAZHUSSAIN
 Microeconomics, Utility, Department of Economics, S. Ajaz Hussain, Sayed Ajaz Hussain

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